This section contains our main results, which relate
weighted Hardy–Sobolev inequalities to the quasiadditivity
of the weighted capacities.
The general approach in the proofs is similar
to those in [1]}, [2]}, [3]}, [4]},
but the present context requires different tools on the level of details.
We first show that the validity of the
\((q,p,\beta )\) -Hardy–Sobolev inequality (REF ) in \(\Omega \) ,
for \(1<p\le q<\infty \) , implies the quasiadditivity property for
the weighted relative capacity
with respect to Whitney balls.
